SCORING rules measure the quality of a probability estimate

Transcription

1 4786 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 10, OCTOBER 2009 Probabilistic Coherence and Proper Scoring Rules Joel B. Predd, Member, IEEE, Robert Seiringer, Elliott H. Lieb, Daniel N. Osherson, H. Vincent Poor, Fellow, IEEE, and Sanjeev R. Kulkarni, Fellow, IEEE Abstract This paper provides self-contained proof of a theorem relating probabilistic coherence of forecasts to their non-domination by rival forecasts with respect to any proper scoring rule. The theorem recapitulates insights achieved by other investigators, and clarifies the connection of coherence and proper scoring rules to Bregman divergence. Index Terms Coherence, probability, scoring rule. I. INTRODUCTION SCORING rules measure the quality of a probability estimate for a given event, with lower scores signifying probabilities that are closer to the event s status ( if it occurs, otherwise). The sum of the scores for estimates of a vector of events is called the penalty for. Consider two potential defects in. There may be rival estimates for whose penalty is guaranteed to be lower than the one for, regardless of which events come to pass. The events in may be related by inclusion or partition, and might violate constraints imposed by the probability calculus (for example, that the estimate for an event not exceed the estimate for any event that includes it). Building on the work of earlier investigators (see below), we show that for a broad class of scoring rules known as proper the two defects are equivalent. An exact statement appears as Theorem 1. To reach it, we first explain key concepts intuitively (the next section) then formally (Section III). Proof of the theorem proceeds via three propositions of independent interest (Section IV). We conclude with generalizations of our results and an open question. Manuscript received March 02, 2008; revised April 21, Current version published September 23, This work was supported by the National Science Foundation (NSF) under Grants PHY to E. H. Lieb, and PHY to R. Seiringer. The work of R. Seiringer was also supported in part by A. P. Sloan Fellowship. The work of D. N. Osherson was supported by the Henry Luce Foundation. The work of H. V. Poor was supported by NSF under Grants ANI and CNS The work of S. R. Kulkarni was supported by the Office of Naval Research under Contracts W911NF and N J. B. Predd is with RAND Corporation, Pittsburgh, PA USA ( jpredd@rand.org). R. Seiringer is with the Department of Physics, Princeton University, Princeton, NJ USA ( rseiring@princeton.edu). E. H. Lieb is with the Departments of Mathematics and Physics, Princeton University, Princeton, NJ USA ( lieb@princeton.edu). D. N. Osherson is with the Department of Psychology, Princeton University, Princeton, NJ USA ( osherson@princeton.edu). H. V. Poor and S. R. Kulkarni are with the Department of Electrical Engineering, Princeton University, Princeton, NJ USA ( poor@ princeton.edu; kulkarni@princeton.edu). Communicated by A. Krzyżak, Associate Editor for Pattern Recognition, Statistical Learning and Inference. Digital Object Identifier /TIT II. INTUITIVE ACCOUNT OF CONCEPTS Imagine that you attribute probabilities and to events and, respectively, where. It subsequently turns out that comes to pass but not. How shall we assess the perspicacity of your two estimates, which may jointly be called a probabilistic forecast? According to one method (due to [1]) truth and falsity are coded by and, and your estimate of the chance of is assigned a score of since did not come true (so your estimate should ideally have been zero). Your estimate for is likewise assigned since it should have been one. The sum of these numbers serves as overall penalty. Let us calculate your expected penalty for (prior to discovering the facts). With probability you expected a score of score of, and with the remaining probability you expected a, hence your overall expectation was. Now suppose that you attempted to improve (lower) this expectation by insincerely announcing as the chance of, even though your real estimate is. Then your expected penalty would be, worse than before. Differential calculus reveals the general fact. Fact: Suppose your probability for an event is, that your announced probability is, and that your penalty is assessed according to the rule: if comes out true; otherwise. Then your expected penalty is uniquely minimized by choosing. Our scoring rule thus encourages sincerity since your interest lies in announcing probabilities that conform to your beliefs. Rules like this are called strictly proper. 1 (We add a continuity condition in our formal treatment, below.) For an example of an improper rule, substitute absolute deviation for squared deviation in the original scheme. According to the new rule, your expected penalty for is whereas it drops to if you fib as before. Consider next the rival forecast of for and for. Because, this forecast is inconsistent with the probability calculus (or incoherent). Table I shows that the original forecast dominates the rival inasmuch as its penalty is lower however the facts play out. This association of incoherence and domination is not an accident. No matter what proper scoring rule is in force, any incoherent forecast can be replaced by a coherent one whose penalty is lower in every possible circumstance; there is no such replacement for a coherent forecast. This 1 For brevity, the term proper will be employed instead of the usual strictly proper /$ IEEE

2 PREDD et al.: PROBABILISTIC COHERENCE AND PROPER SCORING RULES 4787 TABLE I PENALTIES FOR TWO FORECASTS IN ALTERNATIVE POSSIBLE REALITIES fact is formulated as Theorem 1 in the next section. It can be seen as partial vindication of probability as an expression of chance. 2 These ideas have been discussed before, first in [3] which began the investigation of dominated forecasts and probabilistic consistency (called coherence). This work relied on the quadratic scoring rule, introduced above. 3 Reference [5] generalized de Finetti s theorem to a broad class of scoring rules. Specifically, Lindley proved that for every sufficiently regular generalization of the quadratic score, there is a transformation such that a forecast is not dominated by any other forecast with respect to if and only if the transformation of by is probabilistically coherent. It has been suggested to us that Theorem 1 below can be perceived in Lindley s discussion, especially in his Comment 2 [5, p. 7], which deals with scoring rules that he qualifies as proper. We are agreeable to crediting Lindley with the theorem (under somewhat different regularity conditions) but it seems to us that his discussion is clouded by reliance on the transformation to define proper scoring rules and to state the main result. In any event, fresh insight into proper scoring rules comes from relating them to a generalization of metric distance known as Bregman divergence [6]. This relationship was studied in [7], albeit implicitly, and more recently in [8] and [9]. So far as we know, those results have yet to be connected to the issue of dominance. The connection is explored here. More generally, to pull together the threads of earlier discussions, the present work offers a self-contained account of the relations among (i) coherent forecasts, (ii) Bregman divergences, and (iii) domination with respect to proper scoring rules. Only elementary analysis is presupposed. We begin by formalizing the concepts introduced above. 4 III. FRAMEWORK AND MAIN RESULT Let be a nonempty sample space. Subsets of are called events. Let be a vector of events over. We assume that and have been chosen and are now fixed for the remainder of the discussion. We require to have finite 2 The other classic vindication involves sure-loss contracts; see [2]. 3 For analysis of de Finetti s work, see [4]. Note that some authors use the term inadmissible to qualify dominated forecasts. 4 For application of scoring rules to the assessment of opinion, see [9] along with [10, 2.7.2] and references cited there. dimension but otherwise our results hold for any choice of sample space and events. In particular, can be infinite. We rely on the usual notation to denote, respectively, the closed interval, the open interval, and the two-point set containing. Definition 1: Any element of is called a (probability) forecast (for ). A forecast is coherent just in case there is a probability measure over such that for all,. A forecast is thus a list of numbers drawn from the unit interval. They are interpreted as claims about the chances of the corresponding events in. The first event in is assigned the probability given by the first number ( )in, and so forth. A forecast is coherent if it is consistent with some probability measure over. This brings us to scoring rules. In what follows, the numbers and are used to represent falsity and truth, respectively. Definition 2: A function is said to be a proper scoring rule in case (a) is uniquely minimized at for all ; (b) is continuous, meaning that for, for any sequence converging to. For condition 2(a), think of as the probability you have in mind, and as the one you announce. Then is your expected score. Fixing (your genuine belief), the latter expression is a function of the announcement. Proper scoring rules encourage candor by minimizing the expected score exactly when you announce. The continuity condition is consistent with assuming the value. This can only occur for the arguments or, representing categorically mistaken judgment. For if for some, then cannot have a unique minimum at ; similarly, for. An interesting example of an unbounded proper scoring rule [11] is A comparison of alternative rules is offered in [12]. For an event, we let be the characteristic function of ; that is, for all, if and otherwise. Intuitively, reports whether is true or false if Nature chooses. Definition 3: Given proper scoring rule, the penalty based on for forecast and is given by Thus, sums the scores (conceived as penalties) for all the events under consideration. Henceforth, the proper scoring rule is regarded as given and fixed. The theorem below holds for any choice we make. Definition 4: Let a forecast be given. (1)

3 4788 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 10, OCTOBER 2009 (a) is weakly dominated by a forecast in case for all. (b) is strongly dominated by a forecast in case for all. Strong domination by a rival, coherent forecast is the price to be paid for an incoherent forecast. Indeed, we shall prove the following version of Comment 2 in [5]. Theorem 1: Let a forecast be given. (a) If is coherent then it is not weakly dominated by any forecast. (b) If is incoherent then it is strongly dominated by some coherent forecast. Thus, if and are coherent and then neither weakly dominates the other. The theorem follows from three propositions of independent interest, stated in the next section. We close the present section with a corollary. Corollary 1: A forecast is weakly dominated by a forecast if and only if is strongly dominated by a coherent forecast. Proof of Corollary 1: The right-to-left direction is immediate from Definition 4. For the left-to-right direction, suppose forecast is weakly dominated by some. Then by Theorem 1(a), is not coherent. So by Theorem 1(b), is strongly dominated by some coherent forecast. We note that the right side of (2), which is only defined for, can be continuously extended to. This is the content of Lemma 1 in the next section. If the extended satisfies (2) then and (3) Finally, our third proposition concerns a well-known property of Bregman divergences (see, e.g., [13]). When we apply the proposition to the proof of Theorem 1, will be the unit cube in. Definition 6: Let be a convex subset of with nonempty interior. Let be a strictly convex function, differentiable in the interior of, whose gradient extends to a bounded, continuous function on. For, the Bregman divergence corresponding to is given by Because of the strict convexity of, with equality if and only if. Proposition 3: Let be a Bregman divergence, and let be a closed convex subset of.for, there exists a unique, called the projection of onto, such that IV. THREE PROPOSITIONS The first proposition is a characterization of coherence. It is due to [3]. Definition 5: Let. Let the cardinality of be. Let be the convex hull of, i.e., consists of all vectors of form, where,, and. The may be related in various ways, so is possible (indeed, this is the case of interest). Proposition 1: A forecast is coherent if and only if. The next proposition characterizes scoring rules in terms of convex functions. Recall that a convex function on a convex subset of satisfies for all and all, in the subset. Strict convexity means that the inequality is strict unless. Variants of the following fact are proved in [7] [9]. Proposition 2: Let be a proper scoring rule. Then the function defined by is a bounded, continuous, and strictly convex function, differentiable for. Moreover Conversely, if a function satisfies (2), with bounded, strictly convex and differentiable on, and is continuous on, then is a proper scoring rule. (2) Moreover It is worth observing that Proposition 3 also holds if, in which case and (4) is trivially satisfied. V. PROOFS OF PROPOSITIONS 1 3 Proof of Proposition 1: Recall that is the dimension of, and that is the number of elements in. Let be the collection of all nonempty sets of form, where is either or its complement. ( corresponds to the minimal nonempty regions appearing in the Venn diagram of.) It is easy to see that (a) partitions. It is also clear that there is a one-to-one correspondence between and with the property that is mapped to such that for all, iff. (Here, denotes the th component of.) Thus, there are elements in. We enumerate them as, and the corresponding by. Plainly, for all, is the disjoint union of, and hence: (b) For any measure, for all. For the left-to-right direction of the proposition, suppose that forecast is coherent via probability measure. Then for all and hence by (b),. (4)

4 PREDD et al.: PROBABILISTIC COHERENCE AND PROPER SCORING RULES 4789 But the are nonnegative and sum to one by (a), which shows that. For the converse, suppose that, which means that there are nonnegative s, with, such that. Let be some probability measure such that for all. By (a) and the assumption about the, it is clear that such a measure exists. For all Since is minimized at by Definition 2(a), the last term in square brackets is negative. Hence and similarly one shows by (b), thereby exhibiting as coherent. Before giving the proof of Proposition 2, we state and prove the following technical lemma. Lemma 1: Let be bounded, convex and differentiable on. Then the limits and exist, the latter possibly being equal to at or at. Moreover (5) Proof of Lemma 1: Since is convex, the limits exist, and they are finite since is bounded. Moreover, is a monotone increasing function, and hence also exists (but possibly equals at or at ). Finally, (5) follows again from monotonicity of and boundedness of, using that and likewise at. Proof of Proposition 2: Let, let be a proper scoring rule. For (6) By Definition 2(a), the minimum in (6) is achieved at, hence. As a minimum over linear functions, is concave; hence, is convex. Clearly, is bounded (because implies, from (6), that, but a convex function can become unbounded only by going to ). The fact that the minimum is achieved uniquely (Definition 2) implies that is strictly convex for the following reason. We take and and set. Then by uniqueness of the minimizer at. Similarly,. By adding times the first inequality to times the second we obtain, which is precisely the statement of strict convexity. Let.If is differentiable and for all, then (2) is satisfied, as shown by simple algebra. We shall now show that is, in fact, differentiable and. For any and small enough,wehave Since is continuous by Definition 2(b), this shows that is differentiable, and hence. This proves (2). Continuity of up to the boundary of follows from continuity of and Lemma 1. To prove the converse, first note that if is bounded and convex on, it can be extended to a continuous function on, as shown in Lemma 1. Because of strict convexity of we have, for and with equality if and only if. It remains to show that the same is true for. Consider first the case. We have to show that for. By continuity of, (2), and Lemma 1, we have, while.if, the result is immediate. If is finite, we have again by strict convexity of. Likewise, one shows that for. This finishes the proof that is a proper scoring rule. Proof of Proposition 3: For fixed, the function is strictly convex, and hence achieves a unique minimum at a point in the convex, closed set. Let.For,, and hence, by the definition of. Since is differentiable in the first argument, we can divide by and let to obtain The fact that proves the claim. VI. PROOF OF THEOREM 1 The main idea of the proof is more apparent when is bounded. So we consider this case on its own before allowing to reach. Bounded Case Suppose is bounded. In this case, the derivative of the corresponding from (2) in Proposition 2 is continuous and bounded all the way up to the boundary of. (7)

5 4790 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 10, OCTOBER 2009 Let be a forecast and, for, let be the vector with components. Let. Then [Definition 3] [Definition 6] [Proposition 2] [Equation 3] (8) Now assume that is incoherent which, by Proposition 1, means that. According to (4) of Proposition 3, there exists a, namely, the projection of onto, such that for all and hence, in particular, for. Since this proves part (b) of Theorem 1. To prove part (a) first note that weak dominance of by means that for all, by (8). In this case, for all, since depends linearly on.if is coherent, by Proposition 1, and hence. This implies that. Unbounded Case Next, consider the case when is unbounded. In this case, the derivative of the corresponding from Proposition 2 diverges either at or, or at both values, and hence we cannot directly apply Proposition 3. Equation (8) is still valid, with both sides of the equation possibly being. However, if lies either in the interior of, or on a point on the boundary where the derivative of does not diverge, an examination of the proof of Proposition 3 shows that the result still applies, as we show now. If is finite, the minimum of over is uniquely attained at some. Moreover, is necessarily finite. Repeating the argument in the proof of Proposition 3 shows that for any, which is the desired inequality needed in the proof of Theorem 1(b). We are thus left with the case in which lies on an -dimensional face of where the normal derivative diverges. Consider first the case. Then either, in which case is coherent, or or, in which case it is clear that the unique coherent vector strongly dominates. We now proceed by induction on the dimension of the forecast. In the -dimensional hypercube, either lies inside or on a point of the boundary where the normal derivative of is finite, in which case we have just argued that there exists a that is coherent and satisfies for all such that lies in the -dimensional face. In the other case, the induction hypothesis implies that we can find such a. Note that for all the other,. Now simply pick an and choose, where the denote all the elements of outside the -dimensional hypercube. Then for all and also, using Lemma 1,. Hence, we can choose small enough to conclude that for all. This finishes the proof of part (b) in the general case of unbounded. To prove part (a) in the general case, we note that if for and, then necessarily. That is, any coherent is a convex combination of such that. This follows from the fact that a component of can be only if this component is for all the s. The same is true for the value. But the can be infinite only if some component of is and the corresponding one for is,orvice versa. Since for the in question, also by (8) and the assumption that is weakly dominated by. Moreover,. But, hence,. VII. GENERALIZATIONS A. Penalty Functions Theorem 1 holds for a larger class of penalty functions. In fact, one can use different proper scoring rules for every event, and replace (1) by where the are possibly distinct proper scoring rules. In this way, forecasts for some events can be penalized differently than others. The relevant Bregman divergence in this case is given by, where is determined by via (2). Proof of this generalization closely follows the argument given above, so it is omitted. Additionally, by considering more general convex functions our argument generalizes to certain nonadditive penalties. B. Generalized Scoring Rules 1) Non-Uniqueness: If one relaxes the condition of unique minimization in Definition 2(a), a weaker form of Theorem 1 still holds. Namely, for any incoherent forecast there exists a coherent forecast that weakly dominates. Strong dominance will not hold in general, as the example of shows. Proposition 2 also holds in this generalized case, but the function need not be strictly convex. Likewise, Proposition 3 can be generalized to merely convex (not necessarily strictly convex) but in this case the projection need not be unique. Equation (4) remains valid. 2) Discontinuity: A generalization that is more interesting mathematically is to discontinuous scoring rules. Proposition 2 can be generalized to scoring rules that satisfy neither the continuity condition in Definition 2 nor unique minimization. (This is also shown in [9]). Proposition 4: Let satisfy (9)

6 PREDD et al.: PROBABILISTIC COHERENCE AND PROPER SCORING RULES 4791 Then the function defined by is bounded and convex. Moreover, there exists a monotone nondecreasing function, with the property that such that (10) (11) (12) Function is strictly convex if and only if the inequality (9) is strict for. Conversely, if is of the form (12), with bounded and convex and satisfying (10) (11), then satisfies (9). It is a fact [14] that every convex function on is continuous on and has a right and left derivative, and (defined by the right sides of (11) and (10), respectively) at every point (except the endpoints, where it has only a right or left derivative, respectively). Both and are nondecreasing functions, and for all. Except for countably many points,, i.e., is differentiable. Equations (10) (11) say that. The concept of subgradient, well known in convex analysis [15], plays the role of derivative for nondifferentiable convex functions. Note that although and may be discontinuous, the combination is continuous. Hence, if jumps up at a point, has to jump down by an amount proportional to. The proof of Proposition 4 is virtually the same as the proof of Proposition 2, so we omit it. C. Open Question Whether Theorem 1 holds for this generalized notion of a discontinuous scoring rule remains open. The proof of Theorem 1 given here does not extend to the discontinuous case, since for inequality (4) to hold, differentiability of is necessary, in general. REFERENCES [1] G. Brier, Verification of forecasts expressed in terms of probability, Monthly Weather Rev., vol. 78, pp. 1 3, [2] B. Skyrms, Choice & Chance: An Introduction to Inductive Logic. Belmont, CA: Wadsworth, [3] B. de Finetti, Theory of Probability. New York, NY: Wiley, 1974, vol. 1. [4] J. M. Joyce, A nonpragmatic vindication of probabilism, Philosophy of Science, vol. 65, pp , [5] D. V. Lindley, Scoring rules and the inevitability of probability, Int. Statist. Rev., vol. 50, pp. 1 26, [6] L. M. Bregman, The relaxation method of finding a common point of convex sets andits application to the solution of problems in convex programming, U. S. S. R. Comput. Math. Math. Phys., vol. 78, no. 384, pp , [7] L. J. Savage, Elicitation of personal probabilities and expectations, J. Amer. Statist. Assoc., vol. 66, no. 336, pp , [8] A. Banerjee, X. Guo, and H. Wang, On the optimality of conditional expectation as a Bregman predictor, IEEE Trans. Inf. Theory, vol. 51, no. 7, pp , Jul [9] T. Gneiting and A. E. Raftery, Strictly proper scoring rules, prediction, and estimation, J. Amer. Statist. Assoc., vol. 102, no. 477, pp , Mar [10] J. M. Bernardo and A. F. M. Smith, Bayesian Theory. West Sussex, U.K.: Wiley, [11] I. J. Good, Rational decisions, J. Roy. Statist. Soc., vol. 14, pp , [12] R. Selten, Axiomatic characterization of the quadratic scoring rule, Exper. Economics, vol. 1, pp , [13] Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications. Oxford, U.K.: Oxford Univ. Press, [14] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities. Cambridge, U.K.: Cambridge Univ. Press, [15] R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press, Joel B. Predd (S 98 M 01 S 02) received the B.S. degree in electrical engineering from Purdue University, West Lafayette, IN, in 2001, and the M.A. and Ph.D. degrees in electrical engineering from Princeton University, Princeton, NJ, in 2004 and 2006, respectively. He is a policy researcher at the RAND Corporation, Pittsburgh, PA, where his research has addressed topics in information technology, information technology policy, and border security. Prior to joining RAND in 2006, he spent Summer 2004 as a Visiting Researcher at National ICT Australia in Canberra. Robert Seiringer received the Ph.D. degree from the University of Vienna, Vienna, Austria, in He is an Assistant Professor of Physics at Princeton University, Princeton, NJ, where he has been since His research is centred largely on the quantummechanical many-body problem. Dr. Seiringer has been recognized by a Fellowship of the Sloan Foundation, by a U.S. National Science Foundation Early Career award, and by the 2009 Poincare prize of the International Association of Mathematical Physics. Elliott H. Lieb received the B.Sc. degree in physics is from the Massachusetts Institute of Technology (MIT), Cambridge, and the Ph.D. degree in mathematical physics is from the University of Birmingham, Birmingham, U.K. He is a Professor of Mathematics and Higgins Professor of Physics at Princeton University, Princeton, NJ. Previously, he was a Professor at the University of Sierra Leone, Yeshiva University, Northeastern University and MIT. Prof. Lieb is a member of the U.S., Danish, Chilean, and Austrian academies of science and holds honorary Ph.D. degrees from the Universities of Copenhagen, Munich, EPFL Lausanne, and Birmingham. His prizes and awards include the Boris Pregel Award in Chemical Physics of the New York Academy of Sciences, the Heineman Prize in Mathematical Physics of the American Physical Society, the Max-Planck medal of the German Physical Society, the Boltzmann medal in statistical mechanics of the International Union of Pure and Applied Physics, the Schock prize in mathematics of the Swedish Academy of Sciences, the Birkhoff prize in applied mathematics of the American Mathematical Society, the Austrian Medal of Honor for Science and Art, and the Poincaré prize of the International Association of Mathematical Physics. H. Vincent Poor (S 72 M 77 SM 82 F 87) received the Ph.D. degree in electrical engineering and computer science from Princeton University, Princeton, NJ, in From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990, he has been on the faculty at Princeton University, where he is the Dean of Engineering and Applied Science, and the Michael Henry Strater University Professor of Electrical Engineering. His research interests are in the areas of stochastic analysis, statistical signal processing and their applications in wireless networks and related fields. Among his publications in these areas are the recent books MIMO Wireless Communications (Cambridge University Press, 2007), coauthored with Ezio Biglieri et al., and Quickest Detection (Cambridge University Press, 2009), coauthored with Olympia Hadjiliadis.

7 4792 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 10, OCTOBER 2009 Dr. Poor is a member of the National Academy of Engineering, a Fellow of the American Academy of Arts and Sciences, and an International Fellow of the Royal Academy of Engineering, U.K. He is also a Fellow of the Institute of Mathematical Statistics, the Optical Society of America, and other organizations. In 1990, he served as President of the IEEE Information Theory Society, and in , as the Editor-in-Chief of these TRANSACTIONS. He is the recipient of the 2005 IEEE Education Medal. Recent recognition of his work includes the 2007 IEEE Marconi Prize Paper Award, the 2007 Technical Achievement Award of the IEEE Signal Processing Society, and the 2008 Aaron D. Wyner Distinguished Service Award of the IEEE Information Theory Society. Daniel N. Osherson received the Ph.D. degree in psychology from the University of Pennsylvania, Philadelphia, in From 1973 until 1975, he was on the faculty of Stanford University, Stanford, CA, moving to the University of Pennsylvania in From 1978 until 1991, he served as Professor in the Department of Brain and Cognitive Sciences at the Massachusetts Institute of Technology (MIT), Cambridge, and jointly in the Department of Linguistics and Philosophy at MIT from 1986 to From 1991 until 1994, he was Director of the Institut d Intelligence Artificielle in Martigny, Switzerland. From 1995 to 1997, he was Director of the Unité di Razionalité at the Université San Raffaele in Milan, Italy. He then returned to the United States as Professor of Psychology at Rice University, Houston, TX, until 2003 (serving jointly as Professor of Computer Science at Rice from 2000). In 2003, he joined the Department of Psychology at Princeton University, Princeton, NJ, where he is currently a member of the Princeton Neuroscience Institute. His books include Systems that Learn (MIT Press 1986, coauthored with Scott Weinstein, 2nd edition 1999 with coauthors Sanjay Jain, James Royer, and Arun Sharma). Sanjeev R. Kulkarni (M 91 SM 96 F 04) received the B.S. degree in mathematics, the B.S. degree in electrical engineering, the M.S. degree in mathematics from Clarkson University, Potsdam, NY, in 1983, 1984, and 1985, respectively, the M.S. degree in electrical engineering from Stanford University, Stanford, CA, in 1985, and the Ph.D. degree in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge, in From 1985 to 1991, he was a Member of the Technical Staff at MIT Lincoln Laboratory, Lexington, MA. Since 1991, he has been with Princeton University, Princeton, NJ, where he is currently Professor of Electrical Engineering. He spent January 1996 as a research fellow at the Australian National University, 1998 with Susquehanna International Group, and Summer 2001 with Flarion Technologies. His research interests include statistical pattern recognition, nonparametric statistics, learning and adaptive systems, information theory, wireless networks, and image/video processing. Prof. Kulkarni received an ARO Young Investigator Award in 1992, an NSF Young Investigator Award in 1994, and several teaching awards at Princeton University. He has served as an Associate Editor for the IEEE TRANSACTIONS ON INFORMATION THEORY.